Let f : R → R satisfy f(x + y) = 2ˣf(y) + 4ʸf(x), ∀x, y ∈ R. If f(2) = 3, then 14. f'(4)/f'(2) is equal to ____________.

Solution

Maths sir · 4 steps

Swap x and y in the equation; the two versions together force f(x) to be a multiple of 4ˣ - 2ˣ.

1. Swap x and y. Since f(x+y) = f(y+x), write the equation both ways: 2ˣ f(y) + 4ʸ f(x) = 2ʸ f(x) + 4ˣ f(y). Collect the terms to get f(x)(4ʸ - 2ʸ) = f(y)(4ˣ - 2ˣ).
2. Find the form. So f(x)/(4ˣ - 2ˣ) is the same for every x ≠ 0. Call it k, so f(x) = k(4ˣ - 2ˣ). From f(2) = 12k = 3 we get k = 1/4, but k will cancel in the ratio anyway.

   f'(x) = k(4ˣ ln 4 - 2ˣ ln 2) = k ln 2 (2 · 4ˣ - 2ˣ)

3. Put in numbers. f'(4) = k ln 2 (512 - 16) = 496 k ln 2 and f'(2) = k ln 2 (32 - 4) = 28 k ln 2. The ratio is 496/28 = 124/7.
4. So. 14 × 124/7 = 248.

The common slip: 280. Writing ln 4 as ln 2 in the derivative makes the ratio 240/12 = 20 and gives 280. Remember ln 4 = 2 ln 2.

The idea to keep. When a functional equation is not symmetric in x and y, write it again with x and y swapped and equate the two. The function usually falls out directly.

Answer from the NTA final answer key. Solution by PrepWiser.